War arrows were ordered in the thousands for medieval armies and navies, supplied in sheaves normally of 24 arrows.

The White Rose of York was depicted below the branch alluding to the home county of Yorkshire and the sheaves of Sheffield ( Sheaf field ) were shown at either side of the owl's head.

The djed may originally have been a fertility cult related pillar made from reeds or sheaves or a totem from which sheaves of grain were suspended or grain was piled around.

This shutter greatly improved efficiency over the typical Leica shutter by using stronger metal blade sheaves that were " fanned " much faster, vertically along the minor axis of the 24 × 36 mm frame.

Divisions of labour between men, women, and animals that are still in place in Indonesian rice cultivation, were carved into relief friezes on the ninth century Prambanan temples in Central Java: a water buffalo attached to a plough ; women planting seedlings and pounding grain ; and a man carrying sheaves of rice on each end of a pole across his shoulders ( pikulan ).

In 1985 the lift span sheaves, the grooved wheels that guide the counterweight cables, were replaced.

Analogous properties were established by Jean-Pierre Serre ( 1957 ) for coherent sheaves in algebraic geometry, when X is an affine scheme.

Consequently a number of merchant ships were converted to pipe-laying by stripping the interiors and building in large cylindrical steel tanks, fitting special hauling gear and suitable sheaves and guides.

These sheaves were then ' shocked ' into conical stooks, resembling small tipis, to allow the grain to dry for several days before being threshed.

* Expanding collet chucks were used to locate the sheaves by gripping the internal bore, during certain operations.

* Interchangeability of the sheaves and pins was possible, since they were not married to a particular shell.

For example, sheaves were applied to transformation groups ; as an inspiration to homology theory in the form of Borel-Moore homology for locally compact spaces ; to representation theory in the Borel-Bott-Weil theorem ; as well as becoming standard in algebraic geometry and complex manifolds.

Bow sheaves, some very large, were characteristic of all cable ships.

The lyrics were written in 1874 by Knowles Shaw, who was inspired by Psalm 126: 6, " He that goeth forth and weepeth, bearing precious seed, shall doubtless come again with rejoicing, bringing his sheaves with him.

Homological methods and sheaf theory had already been introduced in algebraic geometry by Jean-Pierre Serre and others, after sheaves had been defined by Jean Leray.

Engine power is transmitted via a set of vee-belts that are slack when the engine is idling, but by means of a tensioner pulley can be tightened to increase friction between the belts and the sheaves.

However, this advantage is totally negated by the relatively large energy consumption required to simply move the cable over and under the numerous guide rollers and around the many sheaves.

Using the notion of covering provided by a Grothendieck topology, it becomes possible to define sheaves on a category and their cohomology.

A belt drive is analogous to that of a chain drive, however a belt sheave may be smooth ( devoid of discrete interlocking members as would be found on a chain sprocket, spur gear, or timing belt ) so that the mechanical advantage is approximately given by the ratio of the pitch diameter of the sheaves only, not fixed exactly by the ratio of teeth as with gears and sprockets.

* ' Tohoku ': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept ( to include sheaves of abelian groups ).

A constructible sheaf on a variety over a finite field is called pure of weight β if for all points x the eigenvalues of the Frobenius at x all have absolute value N ( x )< sup > β / 2 </ sup >, and is called mixed of weight ≤ β if it can be written as repeated extensions by pure sheaves with weights ≤ β.

Presheaves and sheaves are typically denoted by capital letters, F being particularly common, presumably for the French word for sheaves, faisceau.

There was perhaps a more direct route available: the abelian category concept had been introduced by Grothendieck in his foundational work on homological algebra, to unify categories of sheaves of abelian groups, and of modules.

It was a possible question to pose, around 1957, about a similar purely category-theoretic characterisation, of categories of sheaves of sets, the case of sheaves of abelian groups having been subsumed by Grothendieck's work ( the Tohoku paper ).

The theory rounded itself out, by establishing that a Grothendieck topos was a category of sheaves, where now the word sheaf had acquired an extended meaning with respect to the idea of Grothendieck topology.

Typically bainite manifiests as aggregates, termed sheaves, of ferrite plates ( sub-units ) separated by retained austenite, martensite or cementite.

Bulk can be reduced by substituting blade sheaves for the plate, but then the rotary FP shutter essentially becomes a regular bladed FP shutter.

Copal collaborated with Nippon Kogaku to improve the Compact Square shutter for the Nikon FM2 ( Japan ) of 1982 by using honeycomb pattern etched titanium foil, stronger and lighter than plain stainless steel, for its blade sheaves.

The lifting hook is operated by the crane operator using electric motors to manipulate wire rope cables through a system of sheaves.

While the role of K above in general Serre duality is played by the determinant line bundle of the cotangent bundle, when V is a manifold, in full generality K cannot merely be a single sheaf in the absence of some hypothesis of non-singularity on V. The formulation in full generality uses a derived category and Ext functors, to allow for the fact that K is now represented by a chain complex of sheaves, namely, the dualizing complex.

) In the case of cohomology of coherent sheaves, Serre showed that one could get a satisfactory theory just by using the Zariski topology of the algebraic variety, and in the case of complex varieties this gives the same cohomology groups ( for coherent sheaves ) as the much finer complex topology.

Siegel was heard to complain that the new theory of functions of several complex variables had few functions in it — meaning that the special function side of the theory was subordinated to sheaves.

As early as 1843, when Europe had been largely at peace for nearly thirty years and most major nations had no plans for war, observers noted sheaves of orders at the Prussian War Ministry, already made out to cover all foreseeable contingencies and requiring only a signature and a date stamp to be put into effect.

This result had been proved previously by Kodaira for the particular case of locally free sheaves on Kähler manifolds.

Cohomology theories have been defined for many different objects such as topological spaces, sheaves, groups, rings, Lie algebras, and C *- algebras.

The arms of nutritionist John Boyd-Orr use two garbs ( wheat sheaves ) as supporters ; the arms of the USS Donald Cook, missiles ; the arms of the state of Rio Grande do Norte in Brazil, trees.

Notice that we did not use the gluing axiom in defining a morphism of sheaves.

In fact it was the need to put ( in particular ) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the formulation of the theory ( with major repercussions for algebraic geometry, in particular from Grauert's work ).

In mathematics, sheaf cohomology is the aspect of sheaf theory, concerned with sheaves of abelian groups, that applies homological algebra to make possible effective calculation of the global sections of a sheaf F. This is the main step, in numerous areas, from sheaf theory as a description of a geometric problem, to its use as a tool capable of calculating dimensions of important geometric invariants.

The first major application was the relative version of Serre's theorem showing that the cohomology of a coherent sheaf on a complete variety is finite dimensional ; Grothendieck's theorem shows that the higher direct images of coherent sheaves under a proper map are coherent ; this reduces to Serre's theorem over a one-point space.

The 39 categories of melakhah are: ploughing earth, sowing, reaping, binding sheaves, threshing, winnowing, selecting, grinding, sifting, kneading, baking, shearing wool, washing wool, beating wool, dyeing wool, spinning, weaving, making two loops, weaving two threads, separating two threads, tying, untying, sewing stitches, tearing, trapping, slaughtering, flaying, tanning, scraping hide, marking hides, cutting hide to shape, writing two or more letters, erasing two or more letters, building, demolishing, extinguishing a fire, kindling a fire, putting the finishing touch on an object and transporting an object between the private domain and the public domain, or for a distance of 4 cubits within the public domain.

* If X is a topological space, then the category of all sheaves of abelian groups on X is an abelian category.

More generally, the category of sheaves of abelian groups on a Grothendieck site is an abelian category.

In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.

Then every Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaves on X.

They exist in several varieties such as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets.

There are also maps ( or morphisms ) from one sheaf to another ; sheaves ( of a specific type, such as sheaves of abelian groups ) with their morphisms on a fixed topological space form a category.

On the other hand, to each continuous map there is associated both a direct image functor, taking sheaves and their morphisms on the domain to sheaves and morphisms on the codomain, and an inverse image functor operating in the opposite direction.

Another example of a presheaf that fails to be a sheaf is the constant presheaf that associates the same fixed set ( or abelian group, or a ring ,...) to each open set: it follows from the gluing property of sheaves that sections on a disjoint union of two open sets is the Cartesian product of the sections over the two open sets.

Let F and G be two sheaves on X with values in the category C. A morphism φ: G → F consists of a morphism φ ( U ): G ( U ) → F ( U ) for each open set U of X, subject to the condition that this morphism is compatible with restrictions.

With this notion of morphism, there is a category of C-valued sheaves on X for any C. The objects are the C-valued sheaves, and the morphisms are morphisms of sheaves.

It can be proved that an isomorphism of sheaves is an isomorphism on each open set U. In other words, φ is an isomorphism if and only if for each U, φ ( U ) is an isomorphism.